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Satellite autonomous orbit determination (OD) is a complex process using filtering method to integrate observation and orbit dynamic equations effectively and estimate the position and velocity of a satellite. Therefore, the filtering method plays an important role in autonomous orbit determination accuracy and time consumption. Extended Kalman filter (EKF), unscented Kalman filter (UKF), and unscented particle filter (UPF) are three widely used filtering methods in satellite autonomous OD, owing to the nonlinearity of satellite orbit dynamic model. The performance of the system based on these three methods is analyzed under different conditions. Simulations show that, under the same condition, the UPF provides the highest OD accuracy but requires the highest computation burden. Conclusions drawn by this study are useful in the design and analysis of autonomous orbit determination system of satellites.

Orbit determination (OD) of satellite plays a significant role in satellite missions, aiming at estimating the ephemeris of a satellite at a chosen epoch accurately. To date, the conventional OD system is dominated by measurements based on (1) ground tracking approaches [

In general, orbit determination is the process of estimating the satellite’s state variables (position and velocity) by comparing (in statistical sense) the difference between the measurement data and the estimated data. Orbit determination system, as shown in Figure

The process of orbit determination.

Owing to the nonlinear dynamic model of satellite orbit motion, the filtering method applied in OD system should be appropriate for nonlinear system [

A variety of autonomous orbit determination methods have been proposed and explored, including a magnetometer-based OD method [

This paper is divided into five sections. After this introduction, the basic descriptions of three filtering methods in autonomous OD system are given in Section

The best known algorithms to solve the problem of autonomous satellite orbit determination are the EKF, UKF, and UPF. In this section, we shall present the theories of the three filter algorithms. These algorithms will be incorporated into the filtering framework based on the dynamic state-space model as follows:

A Kalman filter that linearizes about the current mean and covariance is referred to as an extended Kalman filter or EKF. The EKF is the minimum mean-square-error estimator based on the Taylor series expansion of the nonlinear functions. For example,

The equations for the extended Kalman filter fall into two groups: time update equations and measurement update equations. The specific equations for the time and measurement updates are presented below as shown in (

Predicted state estimate:

Predicted estimate covariance:

The time update equations project the state,

Near-Optimal Kalman gain:

Updated state estimate:

Updated estimate covariance:

The major drawback of EKF is that it only uses the first order terms in the Taylor series expansion. Sometimes it may introduce large estimation errors in a nonlinear system and lead to poor representations of the nonlinear functions and probability distributions of interest. As a result, this filter can diverge [

The unscented Kalman filter (UKF) [

Time = 0, initialize the UKF with

Time =

The equations for the UKF fall into two groups the same as EKF: time update equations and measurement update equations. The specific equations for the time and measurement updates are presented below.

The unscented particle filter (UPF) is a hybrid of the UKF and the particle filter which uses the UKF to get better importance sampling density. A pseudo-code description of UPF is as follows [

(1) Initialization: Time = 0.

Generate

(2) Time =

(a) Update the particles with the UKF:

calculate sigma points from

propagate particle into future by (

incorporate new observation to update the measurement by (

Compute the importance weight

Resampling step:

The basic idea of resampling is to eliminate particles with small weights and to concentrate on particles with large weights. Multiply/suppress particles

Output step:

The overall state estimation and covariance are

The state model (dynamical model) of the celestial OD system for a near-Earth satellite based on the orbital dynamics in the Earth-Centered Inertial (ECI) frame (J2000.0) is

Equation (

The celestial OD method is based on the fact that the position of a celestial body in the inertial frame at a certain time is known and that its position measured in the spacecraft body frame is a function of the satellite’s position. To earth satellite, stars are distributed all over the sky, and the positions of Earth are fixed at a certain time. The geometric relationship among stars, the Earth, and satellite enables us to determine the position of the satellite [

Satellite celestial OD methods can be broadly separated into two major approaches: directly sensing horizon method and indirectly sensing horizon method. In this paper, the directly sensing horizon method is used.

The angle between a star and the earth,

The measurement of celestial OD system.

Assuming a measurement

Geomagnetic OD system relies on measurements from a three-axis magnetometer to determine satellite position and orbit. It uses a model of Earth’s magnetic field and a model of orbital dynamics to predict the time-varying magnitude of Earth’s magnetic field vector at the space. OD system compares the time history of the predicted magnitude and the measured magnitude time history in filter sense to obtain the optimal estimated state (position and velocity) [

Two main models used for describing Earth’s magnetic vector in the geodetic reference frame are World Magnetic Model (WMM) and International Geomagnetic Reference Field (IGRF) [

According to the WMM model 2005, the vector field

This potential

Based on the relationship between magnetic vector, which is obtained by the magnetometer, and the earth magnetic model, the measurement model can be written as

The measurement of geomagnetic OD system.

Assuming a measurement

The trajectory used in the following simulation is a LEO satellite whose orbital parameters are semimajor axis

Figures

Performance of celestial OD system under different sampling intervals.

Sampling interval | RMS (after convergence) | Maximum (after convergence) | |||

Position error/m | Velocity error/m/s | Position error/m | Velocity error/m/s | ||

EKF | 203.318211 | 0.196622 | 532.987272 | 0.486705 | |

UKF | 161.312723 | 0.162900 | 357.545600 | 0.407503 | |

UPF | 159.756079 | 0.160780 | 354.555359 | 0.402378 | |

EKF | 271.640953 | 0.287641 | 703.000581 | 0.613708 | |

UKF | 245.939302 | 0.219864 | 593.829098 | 0.563749 | |

UPF | 245.229683 | 0.219566 | 593.802048 | 0.563396 | |

EKF | 934.238939 | 0.976641 | 2457.895170 | 2.348656 | |

UKF | 736.876288 | 0.699942 | 2322.291896 | 2.302674 | |

UPF | 735.166932 | 0.698808 | 2314.761249 | 2.294924 |

Performance of geomagnetic OD system under different sampling intervals.

Sampling interval | RMS (after convergence) | Maximum (after convergence) | |||

Position error/m | Velocity error/m/s | Position error/m | Velocity error/m/s | ||

EKF | 591.253628 | 0.601946 | 1129.365510 | 1.061735 | |

UKF | 376.894372 | 0.371566 | 877.909403 | 0.799659 | |

UPF | 376.516863 | 0.366538 | 861.513975 | 0.783449 | |

EKF | 1481.673752 | 1.358867 | 2851.660177 | 2.607626 | |

UKF | 705.765450 | 0.648228 | 1325.501267 | 1.276198 | |

UPF | 705.161263 | 0.647876 | 1323.976107 | 1.274585 | |

EKF | 4343.783162 | 4.308921 | 14643.275741 | 11.712547 | |

UKF | 3904.890544 | 3.633798 | 10403.528541 | 10.328902 | |

UPF | 3904.747892 | 3.633460 | 10401.279139 | 10.328554 |

Three filtering methods results of celestial OD system

Three filtering methods results of geomagnetic OD system

The simulations in Figures

This subsection reports how different noise distributions affect the OD performances using three filters. We selected three common noise distributions in navigation, and they are normal distribution, student’s

Figures

Performance of celestial OD system under different noise distributions.

Noise distribution | RMS (after convergence) | Maximum (after convergence) | |||

Position error/m | Velocity error/m/s | Position error/m | Velocity error/m/s | ||

Normal | EKF | 271.640953 | 0.287641 | 703.000581 | 0.613708 |

UKF | 245.939302 | 0.219864 | 593.829098 | 0.563749 | |

UPF | 245.229683 | 0.219566 | 593.802048 | 0.563396 | |

Student’s | EKF | 387.618044 | 0.411669 | 827.358300 | 0.972190 |

UKF | 257.104360 | 0.261611 | 604.325265 | 0.664447 | |

UPF | 246.294197 | 0.250511 | 587.154229 | 0.639735 | |

Uniform | EKF | 385.803609 | 0.398842 | 994.714264 | 0.902353 |

UKF | 350.079516 | 0.339869 | 888.790677 | 0.788940 | |

UPF | 349.537325 | 0.338473 | 880.417206 | 0.782641 |

Performance of geomagnetic OD system under different noise distributions.

Noise distribution | RMS (after convergence) | Maximum (after convergence) | |||

Position error/m | Velocity error/m/s | Position error/m | Velocity error/m/s | ||

Normal | EKF | 1481.673752 | 1.358867 | 2851.660177 | 2.607626 |

UKF | 705.765450 | 0.648228 | 1325.501267 | 1.276198 | |

UPF | 705.161263 | 0.647876 | 1323.976107 | 1.274585 | |

Student’s | EKF | 1059.874849 | 0.828895 | 2900.800178 | 3.184135 |

UKF | 670.321661 | 0.604835 | 1594.060692 | 1.436411 | |

UPF | 669.959004 | 0.604513 | 1593.085809 | 1.436468 | |

Uniform | EKF | 1246.529130 | 1.207728 | 4029.187774 | 3.253167 |

UKF | 776.622486 | 0.767562 | 1985.870577 | 1.781823 | |

UPF | 776.526144 | 0.767452 | 1985.704699 | 1.781594 |

Celestial OD results of three filtering methods under Student’s

Geomagnetic OD results of three filtering methods under Student’s

As the results in Figures

Besides the accuracy, the computation cost is another essential requirement to evaluate the performance of filtering methods. Table

Comparison of computation cost.

Filter | Theoretical value of computation cost | Simulation value of computation cost | ||

Computation time per orbit of celestial OD system(s) | Computation time per orbit of geomagnetic OD system (s) | |||

EKF | 0.3 | 9.5 | ||

UKF | 1.6 | 11.7 | ||

UPF | 39.5 | 313.9 |

The problem of choosing a suitable filtering method for the orbit determination application has been studied here. Three filtering methods for the autonomous orbit determination using either celestial or geomagnetic measurements have been studied and their performances have been compared for the estimation problem.

The algorithms are tested with STK satellite orbit data, and the simulation results demonstrate that UPF yields the best OD accuracy and the EKF yields the worst under the same condition. The main reason is that the state equations and measurement equations for autonomous orbit determination system are significantly nonlinear as well as the non-Gaussian errors.

In addition, the paper analyzed the computation cost of the three filtering methods, and UPF-based OD system can provide the highest OD accuracy, though it requires the largest computation time. However, the UPF can finally meet the real-time requirements, as with the development of computer technology.

The work described in this paper was supported by the National Natural Science Foundation of China (60874095) and Hi-Tech Research and Development Program of China. The authors would like to thank all members of Science and Technology on Inertial Laboratory and Fundamental Science on Novel Inertial Instrument & Navigation System Technology Laboratory, for their useful comments regarding this work effort.